Predicting the Onset of Nonlinear Pharmacokinetics

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1 Citation: CPT Pharmacometrics Syst Pharmacol (208) 7, ; doi:0002/psp4236 ARTICLE Predicting the Onset of Nonlinear Pharmacokinetics Andrew M Stein and Lambertus A Peletier 2 When analyzing the pharmacokinetics (PK) of drugs, one is often faced with concentration C vs time curves, which display a sharp transition at a critical concentration For C >, the curve displays linear clearance and for C < clearance increases in a nonlinear manner as C decreases Often, it is important to choose a high enough dose such that PK remains linear in order to help ensure that continuous target engagement is achieved throughout the duration of therapy In this article, we derive a simple expression for for models involving linear and nonlinear (saturable) clearance, such as Michaelis- Menten and target- mediated drug disposition (TMDD) models CPT Pharmacometrics Syst Pharmacol (208) 7, ; doi:0002/psp4236; published online on 08 September 208 Study Highlights WHAT IS THE CURRENT KNOWLEDGE ON THE TOPIC? Mathematical models of nonlinear PK and TMDD of mabs are widely used to guide drug development Often, it is important to choose a high enough dose such that PK remains linear to help ensure that continual target engagement is achieved throughout the duration of therapy There has not yet been a demonstration for how the PK/ pharmacodynamic parameters impact the onset concentration at which the nonlinearity is observed WHAT QUESTION DID THIS STUDY ADDRESS? How the PK and binding properties of the drug impact the onset of nonlinear PK WHAT DOES THIS STUDY ADD TO OUR KNOWLEDGE? was derived and was found to be equal to /Cl for the Michaelis- Menten model or k syn /(Cl/V c ) for the TMDD model HOW MIGHT THIS CHANGE DRUG DISCOVERY, DEVELOPMENT, AND/OR THERAPEUTICS? The parameter can be used to provide better intuition for how the PK parameters impact drug concentration profiles and, in particular, guide the modeler in understanding where the random effects may need to be added and what parameters of the model are identifiable When analyzing the pharmacokinetics (PK) of drugs, one is often faced with concentration vs time curves which display a sudden increase in the elimination rate below a certain critical concentration ( ) Examples of this phenomenon, in which the PK exhibit this nonlinear behavior, are shown in Figure for efalizumab (anti- CDa), mavrilimumab (antigranulocyte- macrophage colony- stimulating factor receptor), 2 and romosozumab (antisclerostin) 3 This nonlinearity shown in Figure results from a combination of two pathways of drug clearance: (i) a linear, nonspecific clearance due to endocytosis; and (ii) a nonlinear, saturable clearance due to internalization of the target receptor: At high drug concentrations, the target receptor is saturated and the rate of elimination is governed by the nonspecific clearance At lower drug concentrations, when the target receptor is no longer saturated, both the nonspecific route and the saturable route contribute to drug clearance and, hence, the rate of elimination increases as the concentration decreases A common goal when selecting the dose and regimen of a drug is to maintain a drug concentration that stays above the There are two reasons for this goal Ensure lower PK variability in the population, because may vary between subjects 2 Ensure that target occupancy remains high, as maintaining drug concentrations above is thought to be a necessary (although not sufficient) condition to maintain target saturation 4 In this article, we show examples of systems that have a simple critical value for the drug concentration such that as long as the drug concentration remains above this value, target mediated elimination is saturated and the PK is linear In Figure, is illustrated; when the concentration falls below, the elimination rate suddenly increases Novartis Institute for BioMedical Research, Cambridge, MA, USA; 2 Mathematical Institute, Leiden University, RA, Leiden, The Netherlands *Correspondence: Andrew M Stein (andrewstein@novartiscom) Received 2 February 208; accepted 27 April 208; published online on 08 September 208 doi:0002/psp4236

2 67 Figure Nonlinear pharmacokinetics for efalizumab, mavrilumab, and romosozumab Note that for all three drugs, as the concentration drops below the critical concentration ( ), the elimination rate suddenly increases We derive an analytical expression for for both the Michaelis-Menten model,5 and the target- mediated drug disposition model (TMDD) 6 8 Both models have two routes of clearance: nonspecific- linear elimination and saturablenonlinear elimination Both analyses will be discussed in the absence and in the presence of a peripheral compartment In the presence of a peripheral compartment, the analysis becomes more complex, but it is still transparent in common cases, such as rapid exchange between the two compartments In all cases, is independent of drug dose and volume of distribution (a) (b) METHODS For modeling the onset of the PK nonlinearity, we focus on the two models that are frequently used by the pharmacometrics community: Michaelis- Menten and TMDD A more general theoretical derivation of for any PK model is beyond the scope of this article We start by focusing on the simpler case of the one- compartment model and then extend this analysis to the twocompartment scenario The two- compartment models are shown schematically in Figure 2 below The corresponding one- compartment models are identical with the models shown in Figure 2, except that the peripheral compartments (C p ) are removed Michaelis- Menten elimination One- compartment model The basic model for linear and nonlinear drug elimination involves a single (central) compartment and a single differential equation for the concentration C of the drug involving linear and saturable elimination side by side: V c = Cl C V C max C+K M Here, V c is the volume of the central compartment, Cl a first order clearance rate, is the maximum rate of () Figure 2 Structural scheme of the basic model describing (a) Michaelis- Menten kinetics and (b) target- mediated drug disposition (TMDD) where a drug C binds a target receptor R which is synthesized by a zero order reaction and degenerates according to a first order reaction Drug and target form a complex CR, which internalizes through a firstorder reaction saturable elimination and K M is the Michaelis- Menten constant For an initial iv bolus dose D, the initial concentration is given by C(0) = D/V C Two- compartment model When the drug is distributed over a central compartment with concentration C c and a peripheral compartment C p, with linear and nonlinear elimination occurring only from the central compartment, the PK are described by the system of equations: V c c V p p = Cl C c Q(C C c +K c C p ) M in which exchange between the compartments is driven by the difference in drug concentration in the two compartments (ie, by the term Q(C c C p ) where Q denotes a nonspecific clearance rate) C c = +Q(C c C p ) (2) wwwpsp-journalcom

3 672 Table Parameters for mavrilimumab, efalizumab, and romosozumab, based on fits to data Mavrilimumab Efalizumab Romosozumab Units Weight- based dose mg/kg Equivalent molar dose nmol V c L V p L Cl L/d Q L/d k syn = /V c nm/d K ss = K M 2 2 nm k e(r) /d k e(cr) /d k off 0 /d k on = (k off + k e(cr) )/K ss /(nm d) Cl, clearance; k e(c), drug elimination rate; k e(r), receptor elimination rate; K M, Michaelis-Menten constant; k off, dissociation rate; k on, association rate; Kss, quasi-steady-state constant; k syn, receptor synthesis rate; Q, intercompartmental clearance; V c, central volume;, maximal rate of saturable (nonlinear) elimination; V p, peripheral volume If initially, an iv bolus dose D is supplied to the central compartment while the peripheral compartment is empty, the initial conditions are here C c (0) = D/V c and C p (0) = 0 Target- mediated drug disposition In its most elementary form, TMDD target kinetics is assumed to follow a zero- order production rate k syn and a firstorder elimination rate k e(r) R The formation of drug- target complex CR takes place via a second- order process k on C R, a first- order loss of the drug- receptor complex via k off CR, and an irreversible first- order elimination process k e(cr) CR In Figure 2 this system of reactions is shown schematically One- compartment model Mathematically, the onecompartment TMDD model is described by the system of differential equations below: = k onc R+k off CR k e(c) C; k e(c) = Cl v C dr = k syn k on C R+k off CR k e(r) R R = k on C R k off CR k e(cr) CR Note that in the absence of drug, the steady- state target concentration is given by R 0 = k syn /k e(r) Here, we assume that the drug is supplied through an iv bolus dose D to the system when it is free from drug and the target is at steady state (ie, C(0) = D/V c, R(0) = R 0 and CR(0) = 0) By adding the first and the third equations, we obtain an equation for the total amount of drug C tot = C + CR and, similarly, by adding the second and the third equations we obtain an equation for the total amount of the target, R tot = R + CR: tot = k e(c) C k e(cr) CR dr tot = k syn k e(r) R k e(cr) CR Note that the on- rate and the off- rate constants k on and k off no longer appear in these equations (3) (4) In practice, drug binding and complex internalization is fast, so that after a short time drug, target, and drug- target complex are in quasi- steady- state (QSS) 9 (ie, C, R, and CR) are approximately related through the expressions: where K SS C CR = R tot and R = R C+K tot SS C+K SS K SS = k off +k e(cr) k on Two- compartment model The two- compartment model is similar to the one- compartment model with the addition of a peripheral compartment with the initial condition C p (0) = 0 tot ptot = k e(c) C k e(cr) CR Q(C C p ) = +Q(C C p ) dr tot = k syn k e(r) R k e(cr) CR Model analysis, fitting, and simulation A mathematical analysis of the above equations was performed to derive for each model above To test the theory, the data from Figure was fit to the twocompartment TMDD model from Eq 7 where it was assumed that k e(r) = k e(cr) This assumption is typically made when fitting the PK for membrane- bound targets where it can be especially difficult to estimate k e(r) 0 Good fits can still be achieved even when implementing this constraint Sensitivity analyses were performed for the onecompartment Michaelis- Menten and TMDD models and the (5) (6) (7) CPT: Pharmacometrics & Systems Pharmacology

4 673 Concentration (nm), , dose (mg/kg) 00 Vc (L) Vmax (nmol/day) Km (nm) Month CL (L/day) fold change in parameter 0x 3x x 03x 0x Figure 3 Mavrilimumab: one- compartment model in Eq in which the base parameter values are in Table Each plot shows a sensitivity analysis where the parameter in the title is changed from 0- fold to 0- fold The dot shows critical concentration CL, clearance;, maximal rate of saturable elimination two- compartment Michaelis- Menten model In the sensitivity analysis, one parameter at a time was changed while all other parameters were held fixed, and the calculated was plotted together with the simulated PK data Because dosing of monoclonal antibodies (mabs) is usually reported in mg/kg, but the binding model requires doses of nmol and concentrations of nm, a dose of 0 mg/kg is converted to nanomoles for the model simulations using the formula below, which assumes a 70 kg patient and a 50 kda drug (typical antibody) 0 mg kg 70 kg patient g 000 mg mol g 09 nmol = 467 nmol mol RESULTS Model fit The model fits to the data are shown in the Supplementary Material and the parameters from the fits that are subsequently used for simulations are listed in Table These parameters were also used to compute the lines, as described below Note that the large values for k e(r) and k e(cr) for romosozumab and efalizumab are due to the practical unidentifiability of these parameters Michaelis- Menten elimination One- compartment model For drug concentrations, which are either large or small with respect to K M, Eq can be approximated by the following simpler equations: V c = Cl C Cl C for C K M, ( ) V c = Cl + C for C K K M M Thus, on a logarithmic scale, the graph of C(t) will be approximately linear for both large and small values of C; for (8) the range of concentrations in between, the graph curves down connecting the upper linear segment with the lower linear segment (see Figure 3) Of particular interest is the situation when the rate of elimination increases substantially at low concentrations According to Eq 8, this is the case when: Cl K K M M Cl Throughout this article, it will be assumed that Eq 9 holds The sensitivity analysis for romosozumab in the Supplementary Material demonstrates that for small values of or large values of K M, the shoulder (where the onset of the PK nonlinearity is observed) disappears entirely and the PK appears linear As the concentration drops, the elimination rate increases, and it is important to know at which concentration this transition takes place We define this critical concentration,, as the value of C where the rate of elimination doubles from its value at large drug concentrations This definition was chosen because when the rate of elimination doubles, that means the linear and nonlinear components of elimination contribute equally is derived by dividing Eq by C, which gives: ( d V c log (C) = Cl + V ) max C+K M It can readily be seen that for large concentrations, the slope is Cl and, thus, the rate of elimination doubles when: = Cl or C = K () C+K M Cl M By assumption of Eq 9, we may neglect K M in the right expression in Eq and, thus, define the critical concentration to be: (9) (0) wwwpsp-journalcom

5 674 def = Cl (2) Note that neither the drug dose nor the volume V c of the central compartment enters the definition of In Figure 3, we show simulations of concentration graphs based on Eq for data from mavrilimumab in which Cl = 03 L/d, = 67 nmol/d, and K M = nm (Table ) Then, according to Eq 2, = 22 nm Note that here is large compared to K M so that for the base values condition Eq 9 is satisfied Figure 3 shows how, computed from Eq 2 and indicated by a dot, changes when one of the parameters involved in the model is varied In addition, it shows where fits in the concentration- time curve of the drug The graph for the largest parameter value is drawn in red and the graph for the smallest is drawn in magenta In the first panel in Figure 3, in which the dose is varied while keeping C(0) >, it is shown how both numerically and geometrically corresponds to the kink in the graph, irrespective of the dose value According to Eq 2, as increases, moves up, while as Cl increases moves down, and this is observed in the second and third panels Because does not depend on either V c or K M, it does not move in the fourth and the fifth panels Two- compartment model As we shall see, for initial drug concentrations that are large enough, the switch from linear to nonlinear drug elimination is apparent in this situation as well, and it is possible to identify a critical drug concentration Here, we only derive the formula for for the special case when drug exchange between the two compartments is fast relative to the nonspecific clearance (Cl) and the saturable clearance ( ), such as when: Q Cl and Q Under these conditions, it is found that the two concentrations, C c and C p, rapidly coalesce (ie, C c (t) C p (t) 0), and in both the central and peripheral compartments is given by the same formula as for the one- compartment model (ie, by Eq 2): = Cl (3) (4) The details of the derivation of Eq 4 are provided further below In Figure 4, we show simulations for the two- compartment Michaelis- Menten model for parameter values, which are varied around the base parameters listed in Table, as indicated in the headings of the different panels The graphs in the different panels are comparable to those shown in Figure 3 for the one- compartment model, except that many show a weak point of inflection below the Evidently, the location of on the graphs, as defined by Eq 4, has not changed much from what was seen in Figure 3 This is remarkable because the condition in Eq 3 is not satisfied by the base values of the parameters Specifically, the value of the intercompartmental clearance Q is not large compared to On the other hand, Q is indeed large compared to the nonspecific clearance Cl, as is also borne out by the different graphs shown in Figure 4, which exhibit a clear two- phase structure during the initial development Concentration (nm) 0,000, ,000, dose (mg/kg) 00 Vp (L) Vmax (nmol/day) Q (L/day) CL (L/day) Km (nm) Month Vc (L) fold change in parameter 0x 3x x 03x 0x Figure 4 Mavrilimumab two- compartment Michaelis- Menten model, in which the base parameter values are listed in Table, and each plot shows a sensitivity analysis in which the parameter in the title is changed from 0- fold to 0- fold The dot shows critical concentration CL, clearance;, maximal rate of metabolism CPT: Pharmacometrics & Systems Pharmacology

6 675 Rapid exchange between central and peripheral compartments Suppose that Q Cl and Q Then, when we divide the system of Eq 2 by Q, and scale the time variable according to τ = Qt, we obtain, with x (τ) = C c (τ/q) and y (τ) = C p (τ/q), where V dx c = ε x (x y) ν(τ), dτ dy V p dτ = x y ε = Cl Q and ν(τ) = C c Q C c +K M (5) (6) Adding the two equations we find for the total amount of drug: With these equalities, the system in Eq 4 can be approximated by the following pair of equations: Thus, when we subtract the second equation in Eq 23 from the first we obtain for 0 < t < t 0 or, when we multiply by V c, d (C+R tot) = k e(c) C k e(cr) R tot dr tot = k syn k e(cr) R tot = k e(c) C k syn V c = Cl C where = V c k syn (23) (24) (25) the conservation law: A(τ) def = V c x(τ)+v p x(τ), (7) This equation is formally the same as the first equation in the system of Eq 8 Following the reasoning used in the Michaelis- Menten elimination Subsection to derive, we assume, as in Eq 9, that: d dτ (V c x +V p y) = ε x ν 0 (8) Thus, there is very little loss of drug: on the fast time scale τ = O() (ie, t = O(/Q)) Similarly, by dividing the equations by their respective volumes and then subtracting the two equations we obtain: and show that Cl K ss or k syn k e(c) K ss (26) ( d (x y) + ) (x y) dτ V c V p so that C c (t) C p (t) 0 on the fast time scale as well Therefore, within a very short time, we may write A(t) = (V c + V p )C c (t) We now go back to Eq 8 and multiply it by Q so that the original time variable t is returned In light of Eq 9, this yields the following equation: V ss c = Cl C c C c +K M V ss = V c +V p where V ss is the volume of distribution at steady state Following the arguments given for the one- compartment model, we find that the formula for, as given by Eq 2, has not changed Target- mediated drug disposition (9) (20) One- compartment model Suppose that the drug concentration is large compared to the dissociation constant (ie, C(t) K SS over a period 0 < t < t 0 ) Then Eq 5 implies that during this interval we have: CR R tot and R 0 (22) C c (2) = Cl = k syn k e(c) (27) Thus, by assuming rapid drug binding and complex internalization, drug clearance can be seen as the sum of linear and Michaelis- Menten type elimination resulting in the same expression for the In Figure 5, we show simulations for the onecompartment Michaelis- Menten and TMDD models and we see that for mavrilimumab, a typical antibody for which Eq 26 is satisfied, simulations give corresponding results and, as we have seen in Figures 3 and 4 for the Michaelis- Menten model, fits snugly in the arm of the concentration curves In the first five graphs, in which simulations for the Michaelis- Menten model and the TMDD model are shown together (the Michaelis- Menten model is dashed and the TMDD is solid), the Michaelis- Menten model curves lie below the TMDD curves This can be understood by comparing the equations for C tot and R tot in Eqs 4 and 23 and remembering that CR < R tot The similarity between the Michaelis- Menten and TMDD models has been reported elsewhere 0, Comparable findings were observed for the two- compartment model as well Note that the similarity of the Michaelis- Menten and TMDD models may not apply for other biologics, such as bispecific wwwpsp-journalcom

7 676 Concentration (nm), , dose (mg/kg) 00 Kss (nm) 0 0 ksyn (nm/day) ker (/day) CL (L/day) kecr (/day) Vc (L) koff (/day) Month fold change in parameter 0x 3x x 03x 0x model TMDD MM Figure 5 Mavrilimumab one- compartment Michaelis- Menten model (MM; solid line) and target- mediated drug disposition (TMDD; dashed line) model, in which the base parameter values are in Table and each plot shows a sensitivity analysis where the parameters in the title is changed from 0- fold to 0- fold The dot shows critical concentration For the TMDD model, k syn = /V c, Kss = K M target engagers or cytokines where the doses are much lower 7,2 DISCUSSION In practice, is a useful quantity We list a few applications below One application of is in providing an initial estimate for or k syn from a graphical inspection of the data For large, iv doses, Cl can be estimated by noncompartmental analysis by computing the area under the curve (AUC) 0 for a single dose and then computing Cl = Dose/AUC 0- Then, can be estimated from a graph of the PK and Eq 4 is then applied to get = Cl or k syn = Cl/V c Another application is that when fitting a population PK model, there may be patients who have similar PK above (and, thus, similar Cl) but different levels When this is observed, we have found that a random effect on was needed to capture the intersubject variability in the low concentration data Finally, is also useful in understanding the identifiability of TMDD models when fit to data of the type shown in Figure Here, it has been shown that once the linear PK parameters have been identified (say from high- dose data) k syn is the only remaining parameter needed to determine Then, given k syn, K SS is the only remaining parameter needed to determine the slope of the nonlinearity Thus, these two parameters are often estimable from the nonlinear PK alone However, other parameters of the TMDD model, such as the receptor density R 0 = k syn /k e(r) (often assumed to be constant with k e(r) = k e(cr) ) is often not identifiable with PK data alone, as previously reported 0, One key assumption for the estimate to apply is that initially, the drug concentration is large so that the nonlinear elimination term initially has a negligible contribution to the PK curve If this assumption does not hold, there may be no kink and hence no In addition, there may be a kink, but if the dose is low enough such that the nonlinear component already contributes significantly to the elimination, as calculated here will not describe the kink In the extreme case that Cl 0 for a drug with only saturable elimination, and is essentially unobservable Thus as defined here is no longer of practical value in scenarios where the saturable route of elimination dominates for all observable drug concentrations, as is the case for drugs that are given at low doses, such as bispecific target engagers and cytokines To describe the kink in the PK curve in these scenarios (shown in more detail in the Supplementary Material) an alternative definition of would be needed is of its greatest utility for mabs (or other drugs) given at sufficiently high doses, such that both linear and nonlinear elimination phases are observable In summary, when developing antagonists, it is often the goal to pick a dosing regimen where the drug concentration stays above In this article, we have developed a simple formula for this critical concentration: = /Cl = k syn /k e(c) CPT: Pharmacometrics & Systems Pharmacology

8 677 Supplementary Information Supplementary informa tion accompanies this paper on the CPT: Pharmacometrics & Systems Pharmacology website(wwwpsp-journal com) Acknowledgment The authors would like to thank Johan Gabrielsson for many helpful discussions Source of Funding No funding was received for this work Conflict of Interest Andrew Stein is employed by Novartis Institute for BioMedical Research Author Contributions AMS and LAP wrote the manuscript AMS and LAP performed the research AMS and LAP analyzed the data Bauer, RJ, Dedrick, RL, White, ML, Murray, MJ & Garovoy, MR Population pharmacokinetics and pharmacodynamics of the anti- CDa antibody hu24 in human subjects with psoriasis J Pharmacokinet Biopharm 27, (999) 2 Burmester, GR, Feist, E, Sleeman, MA, Wang, B, White, B & Magrini, F Mavrilimumab, a human monoclonal antibody targeting GM- CSF receptor- α, in subjects with rheumatoid arthritis: a randomised, double- blind, placebo- controlled, phase I, first- in- human study Ann Rheum Dis 70, (20) 3 Padhi, D, Jang, G, Stouch, B, Fang, L & Posvar, E Single- dose, placebocontrolled, randomized study of AMG 785, a sclerostin monoclonal antibody J Bone Miner Res 26, 9 26 (20) 4 Cao, Y & Jusko, WJ Incorporating target- mediated drug disposition in a minimal physiologically- based pharmacokinetic model for monoclonal antibodies J Pharmacokinet Pharmacodyn 4, (204) 5 Michaelis, L & Menten, ML The kinetics of the inversion effect Biochem Z 49, (93) 6 Levy, G Pharmacologic target- mediated drug disposition Clin Pharmacol Ther 56, (994) 7 Mager, DE & Jusko, WJ General pharmacokinetic model for drugs exhibiting target- mediated drug disposition J Pharmacokinet Pharmacodyn 28, (200) 8 Mager, DE & Krzyzanski, W Quasi- equilibrium pharmacokinetic model for drugs exhibiting target- mediated drug disposition Pharm Res 22, (2005) 9 Peletier, LA & Gabrielsson, J Dynamics of target- mediated drug disposition: characteristic profiles and parameter identification J Pharmacokinet Pharmacodyn 39, (202) 0 Gibiansky, L, Gibiansky, E, Kakkar, T & Ma, P Approximations of the targetmediated drug disposition model and identifiability of model parameters J Pharmacokinet Pharmacodyn 35, (2008) Stein, AM Practical unidentifiability of receptor density in target mediated drug disposition models can lead to over-interpretation of drug concentration data biorxiv (207) 2 Klinger, M, et al Immunopharmacologic response of patients with B- lineage acute lymphoblastic leukemia to continuous infusion of T- cell engaging CD9/ CD3- bispecific BiTE antibody blinatumomab Blood 9, (202) 208 The Authors CPT: Pharmacometrics & Systems Pharmacology published by Wiley Periodicals, Inc on behalf of the American Society for Clinical Pharmacology and Therapeutics This is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes wwwpsp-journalcom